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Abstract

In this work we discuss and further develop two particular types of complexity reduction techniques: low-rank approximation and reduced basis methods. We will combine adaptive wavelet methods with both reduction techniques.
First, we consider the general question of approximability. We show that eigenfunctions of a class of partial differential equations with a specific structure of the operator are low-rank approximable in a certain sense. Second, we examine the main tool for low-rank approximation: the singular value decomposition. This tool does not apply in Sobolev spaces – the prototypical solution spaces for partial differential equations in variational form. Thus, we investigate extensions of the singular value decomposition in Sobolev spaces. Third, we propose and analyze an adaptive wavelet Galerkin method for high-dimensional elliptic partial differential equations. We complement the theoretical findings with numerical experiments.
Next, we turn to parameter-dependent partial differential equations. We extend classical reduced basis methods by allowing for adaptive snapshot computation in the offline phase. First, we show that the weak greedy algorithm terminates under certain conditions on the adaptive solve accuracy. Second, we introduce a wavelet error estimator for the exact error. Third, we demonstrate the utility and performance of the method with numerical experiments.
We conclude by presenting a speculative idea combining concepts from low-rank approximation, reduced basis and adaptive approximation. We discuss the challenges encountered in realizing this scheme.